A Mathematical Model to Predict the Prevalence and Transmission Dynamics of Tuberculosis in Amansie...

British Journal of Mathematics & Computer Science4(3): 402-425, 2014

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A Mathematical Model to Predict the Prevalence andTransmission Dynamics of Tuberculosis in Amansie West

District, Ghana

I. K. Dontwi1, W. Obeng-Denteh1*, E. A. Andam1 and L. Obiri-Apraku1

1Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi,Ghana.

Received: 30 April 2013Accepted: 14 August 2013

Published: 07 November 2013_______________________________________________________________________

Abstract

In this paper, a Susceptible - Exposed - Infected - Recovered (SEIR) epidemiological model isformulated to determine the transmission of tuberculosis. The equilibrium points of the model arefound and their stability is investigated. By analyzing the model, a threshold parameter R0 wasfound which is the basic reproductive number. It is noted that when R0 < 1 the disease will fail tospread and when R0 > 1 the disease will persist in the population and become endemic. Themodel has two non–negative equilibria namely the disease – free equilibrium and the endemicequilibrium. The graphical solutions of the differential equations were developed using Matlab aswell as the computer simulations.Keywords:Differential equations, exposed and infected, simulation, transmission dynamics,

tuberculosis.

1 Introduction

Tuberculosis, MTB, or TB (short for tubercle bacillus) is a common, and in many cases lethal,infectious disease caused by various strains of mycobacteria, usually Mycobacterium tuberculosis[1]. Tuberculosis typically attacks the lungs, but can also affect other parts of the body. It is spreadthrough the air when people who have an active TB infection cough, sneeze, or otherwise transmittheir saliva through the air [2]. Most infections are asymptomatic and latent, but about one in tenlatent infections eventually progresses to active disease which, if left untreated, kills more than50% of those infected. Tuberculosis (TB) describes an infectious disease that has plagued humanssince the Neolithic times [3].

Physicians in ancient Greece called this illness phthisis to reflect its wasting character. During the17th and 18th centuries, TB caused up to 25% of all deaths in Europe [4].

Original Research Article

British Journal of Mathematics & Computer Science 4(3), 402-425, 2014

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When people with active pulmonary TB cough, sneeze, speak, sing, or spit, they expel infectiousaerosol droplets 0.5 to 5.0 µm in diameter. A single sneeze can release up to 40,000 droplets [5].Each one of these droplets may transmit the disease, since the infectious dose of tuberculosis isvery low, the inhalation of fewer than 10 bacteria may cause an infection [6].

A number of factors make people more susceptible to TB infections. The most important riskfactor globally is HIV; 13% of all TB cases are infected by the virus [7]. This is a particularproblem in sub-Saharan Africa, where rates of HIV are high [8] and [12].

Tuberculosis is closely linked to both overcrowding and malnutrition, making it one of theprincipal diseases of poverty [9]. Those at high risk thus include: people who inject illicit drugs,inhabitants and employees of locales where vulnerable people gather, medically underprivilegedand resource-poor communities, high-risk ethnic minorities, children in close contact with high-risk category patients and health care providers serving these clients [10].

Tuberculosis in Ghana started way back in the pre-independence era when the then colonialgovernment recognized the need to combat the disease due to the threat it posed to the largersociety. In July 1954, the Ghana Society for the Prevention of Tuberculosis was established tosupport and supplement government’s efforts at combating the disease [11].

Over 46,000 new cases of TB annually are estimated by the World Health Organization (WHO) inGhana [12]. However, in reality, less than a third of the estimated number of cases is officiallyreported each year by health facilities in the country. [13] sought to reduce the infectious group bythe reduction of the contact between infectious and exposed individuals.

The first mathematical model of TB was presented by [14]. Following this, there were severalnumerical studies, primarily focusing on cost-effectiveness of different interventions [15,16]. [16]used a model with one progression rate and various latent classes representing different treatmentand control strategies, and argued that vaccination was cost-effective in countries with high TBburdens. Waaler continued his work in 1968 and 1970, [17] and [18], little work on models oftuberculosis appeared in the literature until the mid - 1990’s.

The Amansie West District is one of the 27 districts in the Ashanti region of Ghana. It was carvedout of the Amansie East District of the Ashanti Region in 1989, as part of the Government’sDecentralization Policy. Manso-Nkwanta is the district capital. The district is located in thesouth-western part of the Ashanti Region. It shares common boundary on its Western part withthe Atwima Mponua District. On its northern part can be found the Atwima Nwabiagya and theBosomtwe – Atwima - Kwanhuma Districts, while a regional boundary separates it from Westernand Central Region on its southern part. The predominant occupation is subsistence farming andillegal small scale mining popularly known as Galamsey. Due to the activities of the illegalmining, the district is faced with the problem of immigration making the district exposed toinfectious diseases with tuberculosis being prevalent.

2. Materials and Methods

The mathematical model will be formulated using differential equations based on theepidemiological compartment modelling. The computer software package that will be used tosolve the differential equation model numerically is Matrix Laboratory (Matlab R2010a). Analysis


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and numerical simulations of the model will be conducted. Records showing the trend oftuberculosis cases recorded annually were obtained from the Amansie West District HealthDirectorate, from 2002 - 2011.

In view of the above, the main gap in knowledge to be filled by the paper was to find out the modeof transmission of Tuberculosis and using the SEIR model based on data obtained to analyse itto see whether the disease will be endemic or not when there is an outbreak in the district.

2.1 The Model

We consider the standard SEIR model where the individuals in the population are divided intofour compartments (Fig. 1). The susceptibles (S) which refers to the healthy individuals that hasnot yet come into contact with TB bacterium. The exposed (E) are people who have come intocontact with the disease but are not yet infective or infectious. The infective (I) are those whohave become infected with TB and are able to transmit the disease and the recovered (R) areindividuals who have recovered from TB. The proportions of the individuals in the compartmentof the population, i.e. S, E, I and R, at time t is denoted as S(t), E(t), I(t) and R(t) respectively.

Fig. 1. Flow chart showing the SEIR model

2.2 Model Assumptions

1. Age, sex, social status, race coupled with climatic conditions in the district does notaffect the probability of an individual being infected.

2. The death rate of all individuals is balanced by a birth rate (birth and deaths occurs atequal rates).

3. We assume that an individual is firstly exposed to the disease through interaction withthe infectious but does not become infectious instantly.

4. The disease in transmitted in a closed environment. There is no emigration orimmigration and there is neither birth nor death in the population. Hence the totalpopulation, N of individuals in the district remains constant.

5. Those that are recovered become immuned and are educated about the transmission ofthe disease. The transmission of the disease within the sanatorium is neglected.

6. Lastly, we assume that there is no treatment failure in the sanatorium and therefore apatient will recover or die.

2.3 Model Equations

Applying the assumption here, the total population, N is at full capacity (it value is between 0 andN).

Also, the individuals are likely to be infected by the infectious individuals in case of contactexcept those who are immune. Those who are undetected or late detected infectious are the ones


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contributing to the disease transmission and spread. Those detected are isolated to the hospital forimmediate treatment and education.

The birth and death rate is . Hence N will be the rate at which individuals are born into thesusceptible class without any immunity and S is the rate at which the leave it through death.

The rate at which the susceptible class changes is equal to the rate at which infections occur. Thisoccurs when the disease is passed from an infective individual to a susceptible one. The number ofsusceptible – infective contacts is proportional to the product of ( )S t and ( )I t .

Hence the rate of change in the susceptible individuals is

dS SIN Sdt N

, where SI is rate of infection.

The rate at which individuals leave the susceptible population is equal to the rate at which theyenter the exposed population.

The number of individuals in the exposed class increases since those in the susceptible classbecomes exposed to the disease.

Let the rate which an exposed individual becomes infectious. Then the rate of change of theexposed population is given by

( )dE SI Edt N

Let be the rate at which an infected individual may recover. The rate of change of the infectiveindividuals will be

( )dI E Idt

The rate of change of the recovered is given by

dR I Rdt

This leads to the following formulations of the SEIR model from the description, assumptions andcompartmental diagram and was given as follows:

dS SIN Sdt N

(1)


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( )dE SI Edt N

(2)

( )dI E Idt (3)

dR I Rdt

(4)

The nonlinear system of differential equations formulated above has initial conditions

0 0 0(0) 0, (0) 0, (0) 0S S E E I I and 0(0) 0R R .

Also, the rate of contact, the rate of infection, the recovery rate and the birth and death rate are allnon-negative ( 0, 0, 0 0and respectively).

Hence ( ) ( ) ( ) ( )N t S t E I t R t and

0dS dE dI dRdt dt dt dt (5)

Since 0dNdt and thus N S E I R is a constant.

If we let( ) ( ) ( ) ( )( ), ( ), ( ), , ( )S T E T I t R ts t e t i t and r tN N N N

(6)

Where ( )s t , ( )e t , ( )i t and ( )r t are the respective fractions of the population for susceptible,exposed, infective and recovered individuals (variables and definitions in Table 1).

Then ( ) ( ) ( ) ( ) 1s t e t i t r t

Table 1. Variables and definitions

Variable DefinitionS(t) The number of Susceptible individuals in the population at time, tE(t) The number of Exposed individuals in the population at time, tI(t) The number of Infected individuals in the population at time, tR(t) The number of Recovered individuals in the population at time, tN Total population


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Table 2. Parameters and their definitions

Parameter Definition The rate of contact. It is defined as the average number of effective

contacts with other individuals (susceptible) per infective unit time. The rate at which the exposed individuals become infective or

infectious. The rate at which the infectious individuals recover per unit time. The birth and death rate

The SEIR model is similar to the SIR endemic model except that it has an additional compartmentE or the exposed class. The parameters and their definitions can be found in Table 2. Thiscompartment reflects an inclusion of incubation period. Individuals in this class are infected butnot infectious.

We divide equations (1) and (2) by N and then substitute equation (6) into equations (1) – (4)respectively to obtain

( )ds i sdt

(7)

( )de si edt

(8)

( )di e idt (9)

dr i rdt (10)

With initial conditions 0 0 0 0(0) 0, (0) 0, (0) 0, (0) 0s s e e i i r r

2.4 Basic Reproductive Number, R0

In epidemiology, the basic reproductive number (sometimes basic reproduction rate or ratio) of aninfection is the number of cases one case generates on the average over the course of its infectiousperiod.

The basic reproduction number 0R is the threshold for many epidemiological models. When

0 1R , the infection dies out in the long run (i.e. each infected individual produces one average

less than one new infected individual). One the other hand, if 0 1R , the infection will be able to


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spread in a population (i.e. each infected individual produces more than one new infectedindividual).

For the purpose of our model, we use the approach by [19] to determine the basic reproductivenumber for this thesis and is given by

0 ( )( )R

(11)

Also, the number of contacts between susceptible and infective is given by

(12)

The SEIR model always has at least one solution given by s = 1, and e = i = r = 0. If the thresholdquantity is greater than one then there is a unique endemic equilibrium solution in D where

, , 0, 0, 0, 1D e i r e i r e i r , [20].

2.5 The Herd Immunity Threshold, H1

The herd immunity threshold (H1) is the percentage of the population that needs to be immunizedto control the transmission of the disease. The endemicity of a disease depends on the basicreproduction number, R0. This threshold value can predict whether the disease will approach andspread through the population or not.

The equation given by [21] for estimating the herd immunity threshold is given as

10

11HR

(13)

2.6 Analysis of the Model

In this section, we present the results of stability analysis of the equilibrium points.

2.7 Equilibrium of the Model

The system of equations of the model (Equations 7 - 10) has two non-negative equilibrium points,namely, the disease-free equilibrium, where 0i and endemic equilibrium, where 0i . Theequilibrium point for the disease – free equilibrium is given as:

* * *, , 1,0,0s e i (14)

The characteristic equation about the point (1,0,0) is as follows:


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202 1 0R

where the roots are:

1

20

2

20

3

,

2 2 4 1,

2

2 2 4 1.

2

R

R

It is clear that 1 and 2 are negative. Moreover, 3 is negative when 0 1R and it is positive

when 0 1.R Hence, we deduce that:

If 0 1R , then the disease-free equilibrium, (1,0,0), is locally asymptotically stable.

If 0 1R , then (1,0,0) is unstable.

The endemic equilibrium points for the respective states are as follows:

0 0

0 0

1 11, , , , ,( )e e e

R Rs e i

R R

(15)

2.8 Stability of the Equilibria

To determine the stability of the system, we will consider linearizing the systems of equations (7)– (9) about the equilibrium points by taking the Jacobian of them.

The Jacobian matrix is given by

0( , , )

0

i sJ s e i i s

(16)

For the disease-free equilibrium, we evaluate the Jacobian matrix at the equilibrium points

* * *, , 1.0.0s e i and hence we get


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* * *

0( , , ) 0

0J s e i

(17)

Solving for roots of the characteristic polynomial (with real coefficient) given in the Jacobianmatrix leads to the characteristic equation given as:

3 21 2 3 0a a a (18)

Where

1

2

3

3

2

a

a

a

(19)

Using the Routh – Hurwitz stability criterion analysis we talked about earlier, the conditions

1 0a , 3 0a and 1 2 3 0a a a must hold. If these conditions are true, then all roots of thecharacteristic polynomial equation have negative real part, which concludes that there is a stableequilibrium.

The endemic steady state has equilibrium point given by

0 0

0 0

1 11, , , , ,( )e e e

R Rs e i

R R

The Jacobian matrix of this equilibrium point is evaluated to obtain

0

0

0

( , , ) 1

0

e e e

R

J s e i R

(20)

Solving for the roots of the polynomial in the Jacobian matrix leads to the characteristic equation

3 21 2 3 0b b b (21)


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Where

1 0

2 0

23 0

2(2 )

1 ( )

b Rb R

b R

(22)

From the Routh – Hurwitz stability criterion, we need to determine the stability of thecharacteristic equation above. From the theorem, if the conditions 1 0b , 3 0b and

1 2 3 0b b b are true, then all the roots of the characteristic equation have negative real part,which means a stable equilibrium.

From equation (22), the conditions are true for 0 1R since 1 0b , 3 0b and 1 2 3 0b b b (all non-negative), for all the values of the parameter and 0 1R , hence it is also true. Inconclusion, by the Routh – Hurwitz stability criteria, the endemic steady state is stable when

0 1R .

Also, the endemic steady state is locally asymptotically stable when 0 1R , because

1 31, 1b b and 21 2 3 0 0 2 0.b b b R R b

3. Numerical Analysis and Results

Records obtained from the Amansie West district Health Directorate shows that tuberculosis is anendemic in the district. See Appendix II for the data. For instances, from 2007 to 2011, there hasbeen an increase in the number of cases recorded. In 2007, 34 cases were diagnosed, 2008 (62cases), 2009 (59 cases), 2010 (62 cases) and 2011 (70 cases). Fig. 2 shows the trend oftuberculosis cases recorded in the district from 2002 – 2011.


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Fig. 2. Trend of Tuberculosis cases in the Amansie West District from 2002 – 2011

The table below shows the estimates of the parameters used in the model.

Table 3. Parameters and their estimated values used in the model

Parameter Symbol ValueDeath rate µ 0.00875Transmission rate β 0.5853Infectious rate ɛ 0.1666Recovery rate γ 0.5000

The values of the parameter estimates from Table 3 are substituted into equations (7) – (10) toobtain

0.00875 (0.00875 0.5853 )ds i sdt (23)

05853 0.17535de i edt (24)

0.1666 0.50875di e idt (25)


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05 0.00875dr i rdt (26)

3.1 Estimation of the Basic Reproductive Number, R0

From equation (11), the basic reproduction number for the SEIR model is given by

00.5853 0.1666 1.09305 1

0.00875 0.1666 0.00875 0.5R

Since 0 1R , the prevalence of Tuberculosis will result in an epidemic. This is due to the factthat the rate of transmission is greater than the recovery rate.

The number of contacts between susceptible individuals and the infective ones is calculated fromequation (12) and is given by

0.5853 1.17060.5000

This shows that an average of one tuberculosis patient contacts 1.1706 susceptible individualsduring an infectious period.

3.2 Estimation of the Herd Immunity Threshold, H1

From equation (13), the herd immunity threshold is given as

111 0.0851

1.09305H

Therefore, to control an epidemic, about 8.51% of the population has to be immunized when thereis an outbreak.

3.3 Estimation of the Equilibrium Points

From equation (14), the equilibrium point of the disease – free steady state was determined to be

* * *, , 1,0,0s e i


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The estimates of the endemic steady state equilibrium were determined from equation (15) and isgiven by

0 0

0 0

( 1) ( 1)1, , , ,e e eR R

s e iR R

(27)

1 0.00875(1.09305 1) 0.00875(1.09305 1), ,

1.09305 0.00875 0.1666 1.09305 0.5853 0.914871,0.00042479,0.00139106

3.4 Stability Analysis of the Equilibrium Points

The disease – free equilibrium point for the model was determined as * * *, , 1,0,0s e i . The

stability of the equilibrium point using the Jacobian matrix from equation (17) was determined.Hence, we obtain

* * *

0.00875 0 0.5853, , 0 0.17535 0.5853

0 0.1666 0.50875J s e i

The determination of the roots of the characteristic polynomial equation from the equation (19), isgiven in the Jacobian matrix as follows:

1 3 0.00875 0.1666 0.5 0.69285a

32 0.00875 0.1666 0.00875 0.5 0.5853 0.1666 0.00875 (2 0.00875) 0.1666 0.5 2.3158 10a

53 0.00875 0.00875 0.1666 0.00875 0.5 0.5853 0.1666 7.2639 10a

Note: 3 5 31 2 3 0.69285 2.3158 10 ( 7.2639 10 ) 1.5318 10a a a

Therefore, the characteristic equation is given by

3 2 3 50.69285 2.3158 10 7.2639 10 0 (28)

The results indicates that 1 0a , 2 0a , and 1 2 3 0a a a . From the Routh – Hurwitzstability criterion, these conditions do not hold. Hence, the disease – free equilibrium is anunstable steady state. This means that when an individual infected with mycobacterium


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tuberculosis is present in a susceptible population, it will eventually result in an outbreak of thedisease.

From equation (27), the endemic equilibrium point was given as

, , 0.914871,0.00042479,0.00139106e e es e i . The Jacobian matrix corresponding tothe endemic equilibrium point from equation (20) gives

3

4

9.56141875 10 0 0.53547, , 8.141875 10 0.17535 0.53547

0 0.1666 0.50875e e eJ s e i

The roots of the characteristic polynomial equation given in equation (21) are obtained as follows

1 0.1666 0.5 0.00875(2 1.09305) 0.69366b

32 0.00875 1.09305 2 0.00875 0.1666 0.5 6.5429 10b

2 53 0.00875 1.09305 1 0.00875 0.00875 0.1666 0.5 0.1666 0.5 7.2633 10b

Note: 3 5 31 2 3 0.69366 (6.5429 10 ) (7.2633 10 ) 4.4660 10b b b

The characteristic equation is given by

3 2 3 50.69366 6.5429 10 7.2633 10 0 (29)

The values of the roots of the characteristic polynomial equation satisfy the Routh – Hurwitzstability criterion. This is because 1 0b , 2 0b , and 1 2 3 0b b b .Therefore, the endemicsteady state is asymptotically stable.

3.5 Sensitivity Analysis

Sensitivity analysis deals with the study of how the uncertainty in the output of a mathematicalmodel or system (numerical or otherwise) can be apportioned to different sources of uncertainty inits inputs [22].

The effect on the reproduction number, R0 and the stability of the disease – free equilibrium wasanalysed. This will be done when the value of the parameter β changes whilst µ, ε and γ remainthe same and also, when γ changes whilst µ, β and ε are maintained.


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1. If βis increased and γ is maintained, i.e. β = 0.6853 and γ = 0.5,

0

1

2

3

1 2 3

1.2798 10.69285 0

0.018975 00.000218 0

0.012928 0

Raaaa a a

From these results, the disease-free equilibrium is unstable.

2. If β is reduced and γ is maintained. i.e. β = 0.4853 and γ = 0.5,

0

1

2

3

1 2 3

0.9063 10.69285 00.014344 00.000731 0

0.009207 0

Raaaa a a

Hence, a stable disease – free equilibrium is attained.

From the above results, as the transmission rate increases or the recovery rate decreases, R0 > 1and the disease – free equilibrium is unstable. This indicates that, the disease will spread whenthere is an outbreak. Apparently, as the transmission rate decreases or the recovery rate increases,R0 < 1 and hence the disease – free equilibrium will be stable. This therefore means that thedisease will not spread.

The same scenarios as we did for the disease – free equilibrium.

1. If β increases and γ remains the same. i.e. β = 0.6853 and γ = 0.5,

0

1

2

3

1 2 3

1.2798 10.69529 00.00766 00.000218 0

0.005108 0

Rbbbb b b

The endemic equilibrium is stable.

2. If the β is reduced and γ remains the same. i.e. β =0.4853 and γ = 0.5,


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0

1

2

3

1 2 3

0.9063 10.69203 00.005424 0

0.0000755 00.00382 0

Rbbbb b b

Hence, the endemic equilibrium is unstable.

When the transmission rate increases or the recovery rate decreases, R0 > 1 and hence the endemicequilibrium is stable. This means that the disease will spread when there is an outbreak. On theother hand, as the transmission rate decreases or the recovery rate increases, R0 < 1, and theendemic equilibrium is in an unstable state. This therefore means that the disease will not spread.

3.6 Sensitivity Analysis by Simulation

Numerical simulations on the SEIR model for tuberculosis using the data were done. MatlabR2010a was used and the value of the parameters are found in Table 3. The Matlab codes arefound in the appendix. The effects and the changes that will occur in the model when the values ofeach of the compartments of the model were altered .i.e. Susceptible(S), Exposed(E), Infected(I)and the Removed(R) were looked at. Time was measured in months for a period of one yeardepending on the period of how tuberculosis prevalence occurs. Some assumptions and graphingto see the effect of changes in each compartment of the model.

A total population of 500 individuals in which all belonged to the susceptible was started with.This implies that there were no exposed, infected and recovered individuals in the population. Thesimulation gave the following graphs as shown in Fig. 3.

It can be noticed from the graph that, the number of susceptibles was 500 at the period of timeunder study whiles the number of exposed, infected and recovered are all at a constant number ofzero during the entire period under study. This implies that there will not be any effect on eachcompartment when all the population belong to the susceptible.

Further adjustment of the system was analysed to obtain the results by introducing 10 infected,thereby reducing the susceptibles to 490, no exposed and no recovered individuals into the system.


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Fig. 3. S = 500, E = 0, I = 0, R = 0

It will be observed from Fig. 4 that the number of susceptibles increased with time and becameconstant getting to the end of the period under study. There is also, no effect on the exposedindividuals and the 10 infected reduced with time to zero. The 10 infected moves into therecovered compartment and no infection are recorded again. This may be due to early detectionand probably seeking immunization. This will eventually cause the disease to die the anyway. Byso doing, the recovered individuals begin to increase as the infected recovered and with time, theyall move back to the susceptible compartment.

The final part of the simulation place emphasis on the exposed class. This will help put the diseaseunder control in the district. With the introduction of some exposed individuals into our system,the number of infected tend to decrease but with time, they increase later. The susceptible class


419

was stable at a point in time within the period of study but increase in number afterwards. Whenwe introduce the infected into our system, with no exposed individuals, the disease failed tospread within some short period of time. The introduction of some exposed individuals willincrease the infected after some period of time and hence the disease will spread again. All can befound be in Fig. 5.

Fig. 4. S = 490, E = 0, I = 0, R = 0


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Fig. 5. S = 390, E = 100, I = 10, R = 0

4. Discussion of Results

In this paper, a mathematical model of the prevalence and transmission of tuberculosis in AmansieWest district of the Ashanti Region was studied. The standard SEIR differential equation modelwas used to predict the transmission and spread of tuberculosis. By analyzing the model, we founda threshold parameter, R0, was found which is the basic reproductive number. It was noted thatwhen R0 < 1, then the epidemic will not spread and when R0 > 1, the disease will persist in thepopulation and hence become endemic. The model has two non-negative equilibria, namely, thedisease – free equilibrium and the endemic equilibrium. The existence and stability of the disease– free and endemic equilibria of the model and sensitivity analysis of the model were performed.The Herd immunity threshold, H1, which shows the percentage or proportion of the populationthat needs to be immunized to control the transmission of the disease when there is an outbreak,was also considered.

From the results, the basic reproductive number for the SEIR model was estimated as R0 =1.09305. As indicated, R0 > 1 and this means that, the disease will spread in case there is anoutbreak. Incidentally, an average of one tuberculosis patient contacts 1.1706 susceptibleindividuals during an infectious period.


421

The steady states of the two non – negative equilibria were also found and analyzed. The disease –free equilibrium, (s*, e*, i*) = (1, 0, 0), was found to be unstable which is a saddle point, whilesthe endemic equilibrium, (se, ee, ie) = (0.914871, 0.00042479, 0.00139106), was also found to beasymptotically stable. These equilibrium points and analyses showed that the presence of aninfected individual in a susceptible population will result in an outbreak of the disease.

In the sensitivity analysis, the relationship between the inputs and output variables of the modelwas undertaken. It was checked by looking at how changes in the transmission rate (β) and therecovery rate (γ) affects the endemicity of the disease – free equilibrium and the endemicequilibrium. In this case, the basic reproductive number, R0 was used.

It was seen that, an increase in the transmission rate or a decrease in the recovery rate will makeR0 > 1 and the disease – free equilibrium is unstable. This indicates that, the disease will spreadwhen there is an outbreak. Apparently, as the transmission rate decreased or the recovery rateincreased, the result was R0 < 1and hence the disease – free equilibrium is also asymptoticallystable. This therefore means that the disease will not spread.

Also, when the transmission rate was increased or the recovery rate decreased, the result was R0 >1and hence the endemic equilibrium is stable. This means that the disease will spread when thereis an outbreak. On the other hand, as the transmission rate decreases or the recovery rate increases,R0 < 1, and the endemic equilibrium is in an unstable state. This therefore means that the diseasewill not spread.

It was also found in the model that, increasing the number of infective individuals reduces thenumber of susceptibles whiles the number of exposed individuals also increases. These concludethat, if stakeholders fail to put in place proper measures to control and eradicate the disease, it willspread. Hence, immunization programmes as well as education on tuberculosis must be intensifiedthroughout the communities and the country as a whole so that the disease can be curbed down.

From the paper, the herd immunity threshold was estimated to be 0.0851. This means that, about8.51% of the population need to be immunized in order to control the spread of the disease. This iscan be effective when early detections are reported at tuberculosis clinics for appropriatetreatments. Children should also be vaccinated with Bacillus Calmette–Guérin (BCG) vaccines.This will help effectively disseminate the disease in them. When the percentage of immunedindividuals exceed the herd immunity, the disease will fail to spread. Hence 8.51% shows theminimum percentage of the population that must be screened and immunized on regular basis inorder for the disease not to spread in the district.

From the simulations, it was found out that, an increase in the number of immunized individualsin the population will also increase the level of immunity. This is to say that, if immunization aredone on regular basis, especially among children, it will help increase the number of immunedindividuals thereby decreasing the spread of tuberculosis within the community. From this, we canconclude that all children must be vaccinated to avoid making them exposed, because every childthat is not immunized increases the number of exposed individuals in the system and this puts athreat in the district.


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5. Conclusion

In search for the possible way of eradicating tuberculosis, there is the need to address the issue ofthe mechanism of the transmission and prevalence of the disease.

Many communicable diseases have been modeled using differential equations. The purpose of thiswork was to examine in detail, a mathematical models for the transmission and prevalence oftuberculosis and then to solve them using differential equations. The following assumptions weremade before the model was formulated.

The model also pointed out that early detection has a positive impact on the reduction oftuberculosis transmission; that is there is a need to detect new cases as early as possible so as toprovide early treatment for the disease. More people should be educated in order to createawareness to the disease transmission so that society will be aware of this deadly disease.

Acknowledgement

The authors acknowledge with appreciation the assistance offered by the Amansie West DistrictDirectorate of Health Services for the data they provided on Tuberculosis. The authors are gratefulto the entire staff. They also express their gratitude to unknown reviewers for their usefulcomments.

Competing interests

Authors have declared that no competing interests exist.

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APPENDIX I

M-File for tuberculosis model

function dy=model(t,y,beta,gamma,mu,epsilon)N= 124507;dy=zeros(4,1);dy(1)= mu*N-mu*y(1)-((beta*y(1)*y(3))/N);dy(2)= ((beta*y(1)*y(2))/N)-(mu+epsilon)*y(2);dy(3)= epsilon*y(2)-(mu+gamma)*y(3);dy(4)= gamma*y(3)-mu*y(4);

Scripts Used in Calling the M-File for Tuberculosis Model

mu = 0.00875;epsilon = 0.1666;beta = 0.5853;gamma = 0.5;N= 124705;options = odeset('RelTol',1e-9,'AbsTol',1e-9); [T,Y] = ode45(@emmaseir,[0 12],[390 100 100],options,beta,gamma,epsilon,mu);figure(1)plot(T,Y(:,1),'.')legend('SUSCEPTIBLE')xlabel('Time(Months)');ylabel('POPULATION OF SUSCEPTIBLE');figure(2)plot(T,Y(:,2),'.')legend('EXPOSED')xlabel('Time(Months)');ylabel('POPULATION OF EXPOSED');figure(3)plot(T,Y(:,3),'.')legend('INFECTED')xlabel('Time(Months)');ylabel('POPULATION OF INFECTED');figure(4)plot(T,Y(:,4),'.')legend('REMOVED')xlabel('TIME (MONTHS)');ylabel('POPULATION OF REMOVED');


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APPENDIX II

Amansie West District Health DirectorateTuberculosis (TB) Cases Detected and Recorded

Year Number of cases2002 482003 552004 582005 552006 232007 342008 622009 592010 622011 70

_________________________________________________________________________________© 2014 Dontwi et al.; This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,provided the original work is properly cited.

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